$\int_{}^{}{a^{x} = \frac{a^{x}}{\ln a}}$ ∫tgx = −ln|cosx| ∫ctgx = ln|sinx| $\int_{}^{}{\frac{1}{\cos^{2}x} = \text{tgx}}$ $\int_{}^{}{\frac{1}{\sin^{2}x} = - \text{ctgx}}$ $\int_{}^{}{\frac{1}{x^{2} + a^{2}} = \frac{1}{a}\text{arctg}\frac{x}{a}}$ $\int_{}^{}\frac{1}{x^{2} - a^{2}} = \frac{1}{2a}\ln\left| \frac{x - a}{x + a} \right|$ $\int_{}^{}{\frac{1}{\sqrt{a^{2} - x^{2}}} = \arcsin\frac{x}{a}}$ $\int_{}^{}{\frac{1}{\sqrt{x^{2} - q}} = \ln\left| x + \sqrt{x^{2} - q} \right|}$
$\left( \text{arcsinx} \right)^{'} = \frac{1}{\sqrt{1 - x^{2}}}\text{\ \ \ }\left( \arccos x \right)^{'} = - \frac{1}{\sqrt{1 - x^{2}}}$ $\left( \text{arctgx} \right)^{'} = \frac{1}{x^{2} + 1}\ \ \ (\text{arcctgx})' = - \frac{1}{x^{2} + 1}$
$$ax^{2} + \text{bx} + c = \left\lbrack \left( x + \frac{b}{2a} \right)^{2} - \frac{}{{4a}^{2}} \right\rbrack$$
$$\mathbf{\text{Zbie}}\mathbf{z}\mathbf{\text{ny}}\ \sum_{}^{}a_{n}\ \text{gdy}:\ \operatorname{}\frac{a_{n + 1}}{a_{n}} < 1;\ \ \operatorname{}\sqrt[n]{a_{n}} < 1;\ a_{n} \leq b_{n}\ i\ \sum_{}^{}b_{n} - zbiezny;\ \ calkowy$$
$$\sum_{}^{}\frac{1}{n^{a}} - \left\{ \frac{a > 1 - \text{zbie}z\text{ny}}{a \leq 1 - \text{rozbie}z\text{ny}} \right.\ - \text{\ szereg\ diricHlet}a$$
an∈R: zbieżność bezwzględna: $\sum_{}^{}{|{( - 1)}^{n}a_{n}|} = \sum_{}^{}a_{n}$- zbieżny $${\sum_{}^{}{{( - 1)}^{n + 1}a_{n}}}{\operatorname{}a_{n} = 0\backslash n}{a_{n} \geq a_{n + 1} \geq a_{n + 2} \geq ...}$$
Współrzędne biegunowe (stożki, walce- z=z): r- promień z kątem φ z osią „x” x= r*cosφ y=r*sinφ jakobian= r
Szereg Taylora: $\sum_{n = 0}^{\infty}{f^{\left( n \right)}\left( x_{0} \right)}\frac{\left( x - x_{o} \right)^{n}}{n!} = f\left( x_{0} \right) + f^{'}\left( x_{0} \right)\frac{\left( x - x_{0} \right)^{1}}{1!} + \ \ldots\backslash nR_{n}\left( x \right) = f^{\left( n + 1 \right)}\left( c \right)\frac{\left( x - x_{o} \right)^{n + 1}}{\left( n + 1 \right)!}dla\ c \in \left( x_{0},x \right)lub(x,x_{0})$